Operações básicas (Operações aritiméticas)

Since we'll be using lots of functions in calculus, it helps if we can manipulate functions by themselves, instead of having to write out the definition of every function.

For example, we can perform arithmetic operations on functions. How this works is simple: for any input, first we apply the functions to get intermediate outputs, then we perform the arithmetic operation on the outputs to get the final output.

More specifically, if there are two functions $f$ and $g$ whose codomains are some sort of set of numbers, we can write $f + g$ for a function that outputs $f(x) + g(x)$ for any input $x$.

Similarly, $f - g$ is a function that outputs $f(x) - g(x)$ for any input $x$; $f \cdot g$ or just $fg$ is a function that outputs $f(x) \cdot g(x)$ (the product of $f(x)$ and $g(x)$) for any input $x$; and $f/g$ or is a function that outputs $f(x)/g(x)$ for any input $x$.

That's all there is to it.

The usefulness of this notation may only be seen much later, where we have to describe certain properties of functions that aren't just as simple as "this function produces this output at this input". For example, much later we'll learn about derivatives and how we can perform arithmetic operations on them:

If $f$ and $g$ are differentiable functions, then $(f + g)' = f' + g'$.

It's fine if you don't understand what this means, but compare it to the version where we write out all the functions involved:

If $f$ and $g$ are differentiable functions, then the function $h$ defined by $h(x) = f(x) + g(x)$ satisfies $h'(x) = f'(x) + g'(x)$ for all $x$.

Isn't the first one much simpler?